Coxeter Type Research Articles

Orthogonality relations for deep level Deligne–Lusztig schemes of Coxeter type

Abstract In this paper, we prove some orthogonality relations for representations arising from deep level Deligne–Lusztig schemes of Coxeter type. This generalizes previous results of Lusztig [Lus04], and of Chan and the second author [CI21b]. Applications include the study of smooth representations of p-adic groups in the cohomology of p-adic Deligne–Lusztig spaces and their relation to the local Langlands correspondences. Also, the geometry of deep level Deligne–Lusztig schemes gets accessible, in the spirit of Lusztig’s work [Lus76].

Ribbon 2–knot groups of Coxeter type

Ribbon 2–knot groups of Coxeter type

On the superspecial loci of orthogonal type Shimura varieties

In this note, we study the superspecial loci of orthogonal type Shimura varieties of signature [Formula: see text] with [Formula: see text]. We prove a conjecture of Gross that the superspecial loci and their lift in the integral models can be parametrized by certain homogeneous spaces. As applications, we provide mass formulas for the superspecial loci. We also indicate how Gross’s conjectures can be proved in the case of Coxeter type Shimura varieties using the group theoretic methods of Görtz and He.

Basic loci of Coxeter type with arbitrary parahoric level

AbstractMotivated by the desire to understand the geometry of the basic loci in the reduction of Shimura varieties, we study their “group-theoretic models”—generalized affine Deligne–Lusztig varieties—in cases where they have a particularly nice description. Continuing the work of Görtz and He (2015, Cambridge Journal of Mathematics 3, 323–353) and Görtz, He, and Nie (2019, Peking Mathematical Journal 2, 99–154), we single out the class of cases of Coxeter type, give a characterization in terms of the dimension, and obtain a complete classification. We also discuss known, new, and open cases from the point of view of Shimura varieties/Rapoport–Zink spaces.

On the generation of some Lie-type geometries

Let Xn(K) be a building of Coxeter type Xn=An or Xn=Dn defined over a given division ring K (a field when Xn=Dn). For a non-connected set J of nodes of the diagram Xn, let Γ(K)=GrJ(Xn(K)) be the J-grassmannian of Xn(K). We prove that Γ(K) cannot be generated over any proper sub-division ring K0 of K. As a consequence, the generating rank of Γ(K) is infinite when K is not finitely generated. In particular, if K is the algebraic closure of a finite field of prime order then the generating rank of Gr1,n(An(K)) is infinite, although its embedding rank is either (n+1)2−1 or (n+1)2.

Myosin assembly of smooth muscle: from ribbons and side polarity to a row polar helical model

After decades of debate over the structure of smooth muscle myosin filaments, it is still unclear whether they are helical, as in all other muscle types, or square in shape. In both cases bipolar building units are proposed, but the deduced cross-bridge arrangements are fundamentally different. The opposite polarity of the adjusting longitudinal rows is proposed for the helical structure, while in the case of square filaments, or myosin ribbons, only their two faces are appositively polarized. Analysis of our unpublished archival data on light meromyosin (LMM) paracrystals and myosin rod assemblies as well as the filaments themselves indicated that the rods were assembled with a 6°–7° tilt angle from the rods’ longitudinal axis, in contrast to the lack of tilt in LMM, both exhibiting a 14.3 nm myosin periodicity. Optical diffraction analysis of EM images of the rod assemblies and those of intact myosin confirmed their helical architecture characterized by 28 nm residue translations, 172 nm repeats and 516 nm pitch. A detailed helical model of these filaments was elucidated with bipolar tetramer building units made of two polar trimers. The filaments elongate at their two ends in a head-to-head manner, enabling targeted cross-bridge polarity of the adjacent rows, in the form of a unique Boerdijk–Coxeter type helix, similar to that of collagen or desmin fibers, with the covalent links replaced by a head-to-head clasp.

Classifying right-angled Hecke C*-algebras via K-theoretic invariants

Exploiting the graph product structure and results concerning amalgamated free products of C⁎-algebras we provide an explicit computation of the K-theoretic invariants of right-angled Hecke C⁎-algebras, including concrete algebraic representants of a basis in K-theory. On the way, we show that these Hecke algebras are KK-equivalent with their undeformed counterparts and satisfy the UCT. Our results are applied to study the isomorphism problem for Hecke C⁎-algebras, highlighting the limits of K-theoretic classification, both for varying Coxeter type as well as for fixed Coxeter type.

Dunkl operators for arbitrary finite groups

The Dunkl operators associated with a necessarily finite Coxeter group acting on a Euclidean space are generalized to any finite group using the techniques of non-commutative geometry, as introduced by the authors to view the usual Dunkl operators as covariant derivatives in a quantum principal bundle with a quantum connection. The definitions of Dunkl operators and their corresponding Dunkl connections are generalized to quantum principal bundles over quantum spaces which possess a classical finite structure group. We introduce cyclic Dunkl connections and their cyclic Dunkl operators. Then, we establish a number of interesting properties of these structures, including the characteristic zero-curvature property. Particular attention is given to the example of complex reflection groups, and their naturally generalized siblings called groups of Coxeter type.

A computational technique in Coxeter spectral study of symmetrizable integer Cartan matrices

With any symmetrizable integer Cartan matrix C∈SCarn⊆Mn(Z), a Z-invertible Coxeter matrix CoxC∈Mn(Z) is associated. We study such positive definite matrices up to a strong Gram Z-congruence C≈ZC′ (defined in the paper by means of a Z-invertible matrix B∈Mn(Z)) by means of the Dynkin type DynC, the complex spectrum speccC⊂C of the Coxeter polynomial coxC(t):=det⁡(tE−CoxC)∈Z[t] and the Coxeter type CtypC=(speccC,swC) of C. We show that the strong Gram Z-congruence C≈ZC′ implies the equality CtypC=CtypC′. The inverse implication is an open problem studied in the paper. However, we prove the implication for positive definite symmetrizable integer Cartan matrices C,C′ satisfying any of the following two conditions: (i) if n≤9, the matrices C,C′ are symmetric (i.e., swC=swC′=1) and DynC is one of the simply laced Dynkin graphs An, n≥1, Dn, n≥4, E6, E7, E8, and (ii) n≥2, the matrices C,C′ are not symmetric and DynC is one of the Dynkin signed graphs Bn, Cn, F4, G2. We do it by a reduction to an analogous problem for positive Cox-regular edge-bipartite graphs.Moreover, given n≤9, we present a list of 60 pairwise non-congruent matrices in SCarn such that any positive definite irreducible symmetrizable integer Cartan matrix in Mn(Z), with n≤9, is strongly Gram Z-congruent with a matrix of the list, see Theorem 4.3.

A Coxeter type classification of one-peak principal posets

We present a complete Coxeter spectral classification of finite one-peak posets I=T∪{⁎} that are principal in the sense that their symmetric Gram matrix GI:=12(CI+CItr)∈M|I|(Q) is positive semi-definite of rank |I|−1, where CI∈M|I|(Z) is the incidence matrix of I; equivalently: posets T with the quadratic Tits form qT:Z|T|+1→Z (in the sense of Drozd) non-negative and the kernel KerqT:={v∈Z|T|+1;qT(v)=0}⊆Z|T|+1 infinite cyclic.We continue the study started in M. Gąsiorek, D. Simson (2012) [11]. We show that, up to the strong Gram Z-congruence and the weak Gram Z-congruence, every one-peak principal poset is uniquely determined by its complex Coxeter spectrum speccI⊆C. Moreover, we calculate the exact number of all such posets of a given size (up to isomorphism) and present their Hasse quivers.We finish the paper with conjecture on the exact number of principal (positive) posets. According to our knowledge no formula is known for the number of non-isomorphic posets in a general case.

FINE DELIGNE–LUSZTIG VARIETIES AND ARITHMETIC FUNDAMENTAL LEMMAS

We prove a character formula for some closed fine Deligne–Lusztig varieties. We apply it to compute fixed points for fine Deligne–Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport–Zink spaces arising from the arithmetic Gan–Gross–Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.

A proof of Lusztig's conjectures for affine type G2 with arbitrary parameters

We prove Lusztig's conjectures ${\bf P1}$--${\bf P15}$ for the affine Weyl group of type $\tilde{G}_2$ for all choices of parameters. Our approach to compute Lusztig's $\mathbf{a}$-function is based on the notion of a "balanced system of cell representations" for the Hecke algebra. We show that for arbitrary Coxeter type the existence of balanced system of cell representations is sufficient to compute the $\mathbf{a}$-function and we explicitly construct such a system in type $\tilde{G}_2$ for arbitrary parameters. We then investigate the connection between Kazhdan-Lusztig cells and the Plancherel Theorem in type $\tilde{G}_2$, allowing us to prove ${\bf P1}$ and determine the set of Duflo involutions. From there, the proof of the remaining conjectures follows very naturally, essentially from the combinatorics of Weyl characters of types $G_2$ and $A_1$, along with some explicit computations for the finite cells.

Erratum to: Basic loci of Coxeter type in Shimura varieties

Erratum to: Basic loci of Coxeter type in Shimura varieties

On Coxeter type study of non-negative posets using matrix morsifications and isotropy groups of Dynkin and Euclidean diagrams

We continue our Coxeter type study of a class of finite posets we started in Gąsiorek and Simson (2012). Here we present a more general algorithmic approach to a classification problem for arbitrary posets J, with n≥2 elements, that are positive (resp. non-negative) in the sense that the symmetric matrix CJ+CJtr∈Mn(Z) is positive definite (resp. positive semi-definite), where CJ∈Mn(Z) is the incidence matrix of J. In particular we show that the Coxeter spectral classification of positive (resp. non-negative) posets can be effectively solved using the right action ∗:Mn(Z)×Gl(n,Z)D→Mn(Z), A↦A∗B≔Btr⋅A⋅B, of isotropy groups Gl(n,Z)D of simply laced Dynkin (resp. Euclidean) diagrams D. By applying recent results of the second author in [SIAM J. Discrete Math. 27 (2013) 827–854] we show that, given two connected positive posets I and J with at most 10 points: (i) the incidence matrices CI and CJ of I and J are Z-congruent if and only if the Coxeter spectra of I and J coincide, and (ii) the matrix CI is Z-congruent with its transpose CItr. Analogous results for non-negative posets of corank one or two are also discussed.

Structure and a Coxeter–Dynkin type classification of corank two non-negative posets

Inspired by famous results of Drozd (1974) [13], Nazarova–Roĭter (1975) [42], and Bondarenko–Stepochkina (2005) [8], on the importance of finite posets I with the non-negative Tits quadratic form qˆI:ZI→Z in the representation theory, we study the finite non-negative posets I of corank two, that is, such posets I that the symmetric Gram matrix GI:=12[CI+CItr]∈MI(Q) of I is positive semi-definite of corank two, where CI∈MI(Z) is the incidence matrix of I. A structure of such posets is described in the paper. It is shown that every such a poset I can be obtained in a natural way from a corank zero poset J and a pair of roots of J. Following main ideas of the Coxeter spectral analysis of posets, we associate with any connected corank two poset I the incidence quadratic form qI:ZI→Z, the Coxeter matrix CoxI:=−CI⋅CI−tr, the Coxeter polynomial coxI(t):=det⁡(t⋅E−CoxI)∈Z[t], its complex Coxeter spectrum speccI, the Coxeter number cI∈N∪{∞}, the reduced Coxeter number cˇI∈N, the incidence defect homomorphism ∂˜I:ZI→KerqI⊆ZI, and a simply laced Dynkin diagram DynI. In particular, we study the question when the Coxeter spectrum speccI of I, together with DynI, determines the incidence matrix CI of I (hence the poset I) uniquely, up to a Z-congruence.By applying linear algebra techniques and a combinatorial computer-aided analysis of posets, we prove that the question has an affirmative answer for connected corank two posets I, with at most 15 elements. A complete list of such posets is presented and their Coxeter type invariants are computed. It follows that |I|≥6, there is only one such a poset I of cardinality 6, and 8 such posets of cardinality 7, up to the poset duality I↦Iop. The existence of a ΦI-mesh translation quiver Γ(RI∪KerqI,ΦI) on the set RI∪KerqI is discussed, where RI={v∈ZI;qI(v)=1} is the set of roots of qI.

Applications of matrix morsifications to Coxeter spectral study of loop-free edge-bipartite graphs

Following the spectral graph theory and a Coxeter type technique applied in the representation theory of groups, algebras, coalgebras, and posets, we study the category UBigrn of connected loop-free edge-bipartite graphs Δ with n≥2 vertices (a class of signed graphs) by means of their Coxeter spectrum speccΔ⊂C, i.e., the spectrum of the Coxeter matrix CoxΔ:=−ǦΔ⋅ǦΔ∈Mn(Z)(1.2), where ǦΔ∈Mn(Z) is the non-symmetric Gram matrix (1.1) of Δ. One of our aims is to compute the set CGpoln+ of all polynomials coxΔ(t), with positive graphs Δ in UBigrn, for all n≥2. Moreover, we study the following Coxeter spectral analysis problem (1.4): “Does the Z-congruence Δ≈ZΔ′ hold (i.e. the matrices ǦΔ and ǦΔ′ are Z-congruent), for any pair of connected positive loop-free edge-bipartite graphs Δ, Δ′ such that speccΔ=speccΔ′”. We show that our problems reduce to the study of the set MorD⊆Mn(Z) of matrix morsifications and their mesh root systems, for the simply laced Dynkin diagrams An, Dn, E6, E7, E8. On this way we construct algorithms allowing us to compute the finite set CGpoln+ and we get a solution of the problem (1.4) for n≤9 and Δ,Δ′ of types An and Dn.

Weyl Groups Associated with Affine Reflection Systems of Type $A_1$ (Coxeter Type Defining Relations)

We offer a presentation for the Weyl group of an affine reflection system R of type A_1 as well as a presentation for the so called hyperbolic Weyl group associated with an affine reflection system of type A_1 . Applying these presentations to extended affine Weyl groups, and using a description of the center of the hyperbolic Weyl group, we also give a new finite presentation for an extended ane Weyl group of type A_1 . Our presentation for the (hyperbolic) Weyl group of an affine reflection system of type A_1 is the first nontrivial presentation given in this generality, and can be considered as a model for other types.

Singular Artin monoids of finite coxeter type are automatic

Singular Artin monoids of finite coxeter type are automatic

Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type

In the structure theory of quantized enveloping algebras, the algebra isomorphisms determined by Lusztig led to the first general construction of PBW bases of these algebras. Also, they have important applications to the representation theory of these and related algebras. In the present paper the Drinfel'd double for a class of graded Hopf algebras is investigated. Various quantum algebras, including small quantum groups and multiparameter quantizations of semisimple Lie algebras and of Lie superalgebras, are covered by the given definition. For these Drinfel'd doubles Lusztig maps are defined. It is shown that these maps induce isomorphisms between doubles of bosonizations of Nichols algebras of diagonal type. Further, the obtained isomorphisms satisfy Coxeter type relations in a generalized sense. As an application, the Lusztig isomorphisms are used to give a characterization of Nichols algebras of diagonal type with finite root system.

On the coincidence of two manifolds associated with the Calogero model

Let R be a root system, for example, the root system associated to a semisimple Lie algebra. In [1], V. A. Golubeva and V. P. Lexin constructed two algebraic manifolds (Bethe and Dunkl manifolds) using the "universal" Dunkl operators. These manifolds were defined as subsets of the complex space ? N of dimension equal to the number of roots of the root system under consideration. The first manifold (Bethe manifold) is characterized by the following property: the Laplace operator constructed by means of Dunkl operators coincides with the "universal" Hamiltonian of the Calogero model. The second one (Dunkl manifold) is characterized by the property: the "universal" Dunkl operators commute. In this paper, the manifolds associated with the irreducible root system of Coxeter type are considered. We give their construction supposing that these manifolds are embedded in ? N/2. A theoremon the coincidence of Bethe and Dunkl manifolds is proved.